Calculation and precision processing of cardiocle and expanded cardioid casing curved surfaces for eccentric rotor vane pumps

ABSTRACT

This invention includes the derivation of the exact mathematical expressions for the curvature, either cardiocle or expanded cardioid, of the casing of the springless eccentric rotor vane pump, thereby facilitating the precision manufacture of the curved surfaces of the casing using CNC techniques. As a result, the capacity and accuracy of the eccentric rotor vane pump is greatly improved. As the section manufacture and assembly of the casing becomes possible, the mass production of large-sized pumps of 1-meter or larger diameter is now attainable, hitherto regarded as almost impossible, and therefore production cost is also reduced. The unique design which positions the axis of eccentricity in the lower central part of the axis of rotor rotation results in increase in the rotation speed of the rotor, and leads to reduction of friction between the vane ends and the curved surface of the casing as the weight of the vane does not affect the movement of the rotor.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention describes the precision processing of curved surfaces of the cardiocle and expanded cardioid casing in springless eccentric rotor vane pumps.

2. Description of the Prior Art

In general, vanes used in eccentric rotor vane pumps are fitted with springs so that their length can vary in line with casing surfaces. However, the eccentric rotor vane pump discussed here has a solid vane of constant length. For this type of eccentric rotor vane pump, the key technology is the accuracy of the casing surface curvatures, to allow the edges of a sliding vane match the surface curves as closely as possible no matter what the rotation angle and the eccentricity of the rotor may be.

However, the exact mathematical descriptions which accurately represent the curves drawn by the movements of the vane edges in an eccentric rotor vane pump have not been found until now. Thus processing of curved casing surfaces has been possible only via the recopy method. This method has several significant weaknesses: (1) Curved surfaces have to be retraced and remodelled each time eccentricity or casing size needs to be changed. (2) Precision processing is not quite possible, especially for large-sized casings. (3) The entire surface of the casing has to be processed at one time. (4) The edges of scraping, sliding vanes make poor contact with casing surfaces.

Moreover, with this recopy method, the accuracy of casing surface processing varies with the eccentricity of the pump, the angle of rotation of the vane, and the distance the vane travels. As there have been no geometrical equations which exactly describe the curves drawn by the vane rotation, such advanced manufacturing techniques as CNC, and processing in sections, have not been available. The only possible manufacturing method was the recopy method, using a prototype curved action.

SUMMARY OF THE INVENTION

In this invention, however, the following equations (A) and (B), which represent the curves drawn by the movement of vanes of fixed length in eccentric rotor vane pumps, are derived on the basis of these curves always falling into two categories, cardiocle and expanded cardioid curves, regardless of rotor eccentricity and vane length: $\begin{matrix} {P = {2a\left\{ {1 + {\left( {R - r} \right)\frac{\sin \quad \theta}{2a}} - \frac{\sqrt{R^{2} - {\left( {R - r} \right)^{2}\quad \cos^{2}\theta}}}{2a}} \right\} {for}{\quad \quad}{cardiocles}}} & (A) \end{matrix}$

$\begin{matrix} {P = {2a\left\{ {1 + {\left( {R - r} \right)\frac{\sin \quad \theta}{2a}}} \right\} \quad {for}{\quad \quad}{expanded}\quad {cardioids}}} & (B) \end{matrix}$

Nomenclature in the equations will be discussed in detail later, in reference to FIGS. 1, 3, 5 and 6.

These two equations represent in terms of analytic geometry the curved surfaces of eccentric rotor pump casings, and thereby alow the precision processing of casings using CNC techniques. As the equations do not depend on rotor eccentricity and vane length, casings of any size can be manufactured to the highest levels of accuracy current engineering technology permits; and even further, more processing in sections is now possible.

As a result, not only precision processing, but also mass production, of large-sized springless eccentric rotor vane pumps of 1-meter or larger diameter is now possible, thus making feasible the supply to customers of eccentric rotor vane pumps at more reasonable prices.

In other current eccentric rotor vane pumps, the center of eccentricity of the rotor is set at the upper section or sides of the casing center for better ventilation and smooth valve movement. But the movement of a vane causes friction with the casing surfaces, as the centrifugal force generated by the rotating vane is in the same direction as the gravitation force exerted on the rotor. Therefore the rotation speed of the rotor has to be kept low. However, the vane of the eccentric rotor vane pump being discussed here makes large-area contact with the casing surfaces when sliding on surfaces; and thus the center of eccentricity of the rotor can be placed in the lower section of the casing center. Additionally, the centrifugal force produced by the rotation of the vane is reduced by the weight of the vane. Therefore the rotation speed of the rotor can be sped up.

In particular, as shown in FIG. 10, existing thrust bearings may be used for the processing of large-sized casings of 1-meter or greater diameter, so that the rotor axis can be designed vertically, reducing gravitational pull due to the weight of the rotating vane and increasing operational life.

As the casing diameter increases, the weight of the vane increases and so, too, does the friction produced by the vane when sliding and scraping along the casing surface. For this reason the manufacture of large-sized eccentric rotor vane pumps was regarded as impractical in the past.

By positioning the rotor shaft vertically, it is possible to reduce the friction between the ends of the vane and the casing surface, and thus to increase the size of eccentric rotor pumps. Furthermore the mathematical descriptions of cardiocle and expanded cardioid curves derived and shown in this invention allows the implementation of CNC techniques in the manufacture of casings, and subsequent increase in casing surface accuracy. CNC processing makes possible both mass production and cost reduction.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a geometric representation of the movement of an eccentric rotor as contained in the invention referred to in this invention.

FIG. 2 compares a cardiocle with a simple cardioid.

FIG. 3 shows the operation of an eccentric rotor vane pump with a cardiocle casing.

FIG. 4 is the actual description of an eccentric rotor vane pump with a cardiocle casing.

FIG. 5 compares the curvatures of cardiocle and expanded cardioid casings.

FIG. 6 shows the relationship between the size of an eccentric rotor and an expanded cardioid.

FIG. 7 shows the operation of an eccentric rotor vane pump with an expanded cardioid casing.

FIG. 8 describes section processing of a pump casing using the methodology introduced in this invention.

FIG. 9 describes an eccentric rotor vane pump of horizontal design.

FIG. 10 describes an eccentric rotor vane pump of vertical design.

FIG. 11 displays the components of the eccentric rotor vane pump described in this invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The derivation of the two equations for cardiocles and expanded cardioids, in reference to the figures and in terms of analytic geometry, are shown below.

FIG. 1 shows a cross-section of an eccentric rotor pump in Cartesian coordinates, for geometric analysis of the casing surfaces of the pump. The surface of circular rotor {circle around (2)} touches basic circle {circle around (1)} at point internally Ĉ. When rotor {circle around (2)} rotates anticlockwise by θ° around the axis of eccentricity, which goes through point Oe, vane {circle around (3)}, which is inserted in rotor {circle around (2)}, also rotates in the same direction as the vane, sliding and scraping along the casing surface. One end of vane {circle around (3)}, P₁ (X₁, Y₁), then moves along the arc of basic circle {circle around (1)}, i.e. J₁→Ĉ→J₂. Vane {circle around (3)} moves in the direction of the diameter along the two guides between the two crescent halves of the assembled rotor {circle around (2)}, passing through the eccentricity center Oe. The other end, P₂ (X₂, Y₂), describes the dotted curve {circle around (4)}.

The length of vane {circle around (3)} is constant; ie., the distance between P₁ (X₁, Y₁) and P₂ (X₂, Y₂), 2{square root over (r+L (2R −r+L ))}=2a, is also constant. This means that the distance between the two points J₁ and J₂ on the x-axis, and the distance between the two points on the y-axis, Ĉ of the perigee and {circle around (m)} of the apogee, are constant. Here, an idealized curve {circle around (4)} is produced, where the distance between any two points on the curve passing through the center is always constant. If the radius of basic circle {circle around (1)}, R, and the radius of rotor {circle around (2)}, r, are determined, a mathematical equation describing the motion of the two ends of vane {circle around (3)}, P₁ and P₂, can be derived, with the angle of rotation, θ°, as the only variable.

Then the equation which describes the curve {circle around (4)} is written in Cartesian coordinates as:

X²+Y² ={2{square root over (r+L (2R−r+L ))}+( R−r)sin θ−{square root over (R ²+L −(R−r+L )² +L cos²+L θ)}}²  (1),

where 0°≦θ≦180°.

In this equation, r denotes the radius of rotor {circle around (2)}, R denotes the radius of basic circle {circle around (1)}, and θ is the angle of rotation of vane {circle around (3)}. This equation, in polar coordinates, is:

P=2{square root over (r+L (2R−r+L ))}+( R−r)sin θ−{square root over (R ²+L −(R−r+L )²+L cos²+L θ)}  (2)

The equation describing the basic circle {circle around (1)} can be written as:

X²+Y²={{square root over (R ²+L −(R−r+L )² +L cos²+L θ)}−(R−r)sin θ}²  (3)

in Cartesian coordinates, and

P={square root over (R²+L −(R−r+L )²+L cos²+L θ)}−( R−r)sin θ  (4)

in polar coordinates.

If half of the length of the vane, {square root over (r(2+L R−r),)} is replaced with a into Equations (1) or (2), the equation becomes: $\begin{matrix} {P = {2a\left\{ {1 + {\left( {R - r} \right)\frac{\sin \quad \overset{.}{\theta}}{2a}} - \frac{\sqrt{R^{2} - {\left( {R - r} \right)^{2}\quad \cos^{2}\theta}}}{2a}} \right\}}} & (5) \end{matrix}$

This equation is equivalent to Equations (2) and (4) for curve {circle around (1)} and {circle around (4)}, i.e., the equation for cardiocles. Equation (5) resembles the equation for a simple cardioid, P=a(1+sin θ), for dotted curve 4′ in FIG. 2. But, Equation (5) is smaller by its third term, {square root over (R²+L −(R−r+L )² +L cos²+L θ)}, than that describing curve 4′. In other words, equation (5) shows a curve 4′ as a cardioid flattened by the amount {square root over (R ²+L −(R−r+L )² +L cos²+L θ)} in comparison with an ordinary cardioid 4′ in the range, 0°≦θ≦180°. And this cardioid curve connects at the two points J₁ and J₂ with the arc of circle {circle around (1)} in the range 180°≦θ≦360°. This composite curve describes the curve drawn by the full rotation of vane {circle around (3)}. It is named “cardiocle” for being a flattened cardioid in the range, 0°≦θ≦180°, and for being a circle in the range, 180°≦θ≦360°.

FIG. 2 gives graphical comparison of the composite cardiocle curve {circle around (4)} with an ordinary cardioid 4′, calculated and drawn using a computer in accordance with the widely-known cardioid equation and the cardiocle equation (5) derived here. As shown in FIG. 2, the distance between the y-intercept of the cardioid 4′ and the lower point Oe is 2a=2r{square root over (r(2+L R−r))}; and thus dotted cardiocle curve {circle around (4)} is the flattened down by r, the radius of the rotor {circle around (2)}, along the y-axis in the range y≧0; and expanded below Oe, also by the amount r. Along the y-axis in the range of yso--; Curve {circle around (4)}, a cardiocle, has the composition of a cardioid in the J₁-m-J₂ section and of a circular arc in the J₁-C-J₂ section.

FIG. 3 is a mechanical drawing, which describes the movement of an eccentric rotor pump with a cardiocle casing. An exact equation, in which the only variable is θ, the angle of rotation of vane {circle around (3)} or rotor {circle around (2)}, can be derived to represent the above-mentioned cardiocle curve drawn by rotation of the vane. Using this equation, accurate casing surfaces can not be processed through CNC techniques.

As shown in FIGS. 3 and 4, the casing is fitted with an inlet, {circle around (13)}, and an outlet, {circle around (14)}, for the flow of liquid into and out of the pump. The inlet and outlet are shown in the fourth and third quadrangles in FIGS. 3. The outer periphery of the casing is surrounded by a cooling chamber, to the outer side of which water jackets are attached.

When the vane mounted on the rotor, as in FIGS. 3 and 4, is rotated anticlockwise, suction force is produced in the casing section containing inlet {circle around (13)}, due to pressure decrease, and drainage force in the section containing outlet {circle around (14)}, due to pressure increase. Fluid inflow and outflow are repeated in tandem with the rotation of the rotor.

In addition to the heat generated by friction between rotating rotor {circle around (2)} and vane {circle around (3)} and the casing surface {circle around (4)}, additional heat is generated due to the continuous kinetic movement of fluid molecules during the repeated inflow and outflow of the liquid. This problem can be solved by applying current water-cooling or air-cooling techniques. Other current eccentric rotor vane pumps require substantial amounts of high-viscosity sealing oil, as their vane ends do not closely or uniformly scrape along tne casing surfaces due to their inaccurately processed casings. However, the equations for curve {circle around (4)} derived in this invention make possible the processing of casing surfaces to the highest possible degree of accuracy, thus requiring only small amounts of low-viscosity sealing oil and making operations more economical.

In order to acquire different curvatures, a curve was drawn using Equation (5) minus the last term, {square root over (R²+L −(R−r)² +L cos²+L θ)}. This new curve also shows that the length of the vane, or casing diameter, remains constant during full rotations. From this, a new equation (6), for what we will call an “expanded cardioid” from now on, is derived. $\begin{matrix} {{P = {2a\left\{ {1 + {\frac{\left( {R - r} \right)}{2a}\sin \quad \theta}} \right\}}}\quad} & (6) \end{matrix}$

This new equation is represented by curve 4″ in FIG. 5. This curve is not defined as an ellipse by mathematical definition, although it looks like one. Equation (6) shows that it is an expanded form of the ordinary cardioid (P=a(1+sin θ)); and is thus named an “expanded cardioid”. As shown in FIG. 5, the expanded cardioid curve 4″ is an enlargement, by R the radius of basic circle {circle around (1)}, of the cardiocle curve {circle around (4)}, in both directions along the y-axis. The length of the vane for this curve, as shown in FIG. 6, is exactly twice that for the cardiocles as shown in FIGS. 1 and 2. This equation can be effectively and ideally applied in the precision processing of another type of eccentric rotor vane pump with expanded cardioid casing. As this expanded cardioid curve is closer to a circle than a cardiocle, rotor movement is expected to be smoother.

In the case of the expanded cardioid curve 4″ shown in FIG. 6, the radius of the rotor is 2{square root over (r(2+L R−r))}−R+r. The rotor is positioned symmetrically, (2{square root over (r(2+L R−r))}−R+r) above the lower y-intercept and (2{square root over (r(2+L R−r))}+R−r) below the upper y-intercept, on the y-axis. Thus the center of the rotor can be exactly determined.

An interesting comparison can be made here; Equation (6) for the expanded cardioid suffices for the range 0°≦θ≦360°, while Equation(5) for the cardiocle suffices only for the range 0≦θ≦180°.

The equations (1) through (6) derived in this invention form a mathematical basis for computer numerical controlled manufacturing of casings of eccentric rotor vane pumps. On the basis of these equations, part processing and assembly of casings of sizes far surpassing the limits set by currently available machine tool technology is now possible for any R and r, the respective radii of any arbitrary primary circle and any eccentric rotor. As CNC techniques become used instead of the tradtional recopy method, mass production becomes possible, thus reducing production costs and allowing the production good quality pumps at reasonable prices. Furthermore, as manufacturing in sections becomes possible, no additional processing equipment is required for large-size casings.

As one practical example of this, invention, FIG. 7 illustrates the operation of a springless eccentric rotor vane pump with an expanded cardioid casing. FIG. 8 describes section processing of a pump casing where the radius R of the basic circle {circle around (1)} is 1,000 mm and the radius r of the eccentric rotor {circle around (2)} is 600 mm. The shaded areas in sectors A, B and C are the parts to be processed in sections using the methodology introduced in this invention. The following table 1 shows the coordinates (x, y) calculated with the equations which describe the two-dimensional cross section of the casing (FIG. 8), over the range 0≦θ≦90°.

TABLE 1 X Y 0° ≦ θ ≦ 30° 0.327692 0.120531 0.655831 0.240369 0.984415 0.359512 1.313441 0.477958 1.642910 0.595706 1.972818 0.712755 2.303164 0.829101 2.633947 0.944745 2.965165 1.059685 3.296817 1.173918 3.628899 1.287444 3.961412 1.400260 4.294353 1.512366 4.627720 1.623759 4.961512 1.734439 5.295727 1.844403 5.630364 1.953650 5.965420 2.062180 6.300894 2.169989 6.636784 2.277076 6.973088 2.383441 7.309805 2.489081 7.646933 2.593996 7.984470 2.698183 8.322415 2.801641 8.660765 2.904369 8.999519 3.006365 9.338676 3.107628 9.678233 3.208156 10.018189 3.307948 10.358541 3.407003 10.699289 3.505318 11.040429 3.602893 11.381962 3.699726 11.723883 3.795816 12.066193 3.891162 12.408889 3.985761 12.751969 4.079612 13.095432 4.172715 13.439275 4.265068 13.783497 4.356669 14.128096 4.447516 14.473070 4.537610 14.818417 4.626948 15.164135 4.715528 15.510223 4.803350 15.856679 4.890413 16.203500 4.976714 16.550685 5.062252 16.898233 5.147027 17.246140 5.231037 17.594406 5.314281 17.943028 5.396756 18.292005 5.478463 18.641335 5.559399 18.991015 5.639564 19.341044 5.718956 19.691420 5.797573 20.042140 5.875415 20.393204 5.952481 20.744610 6.028768 21.096354 6.104277 21.448436 6.179005 21.800853 6.252951 22.153604 6.326114 22.506686 6.398494 22.860097 6.470088 23.213837 6.540895 23.567901 6.610914 23.922290 6.680145 24.277000 6.748586 24.632031 6.816235 24.987379 6.883092 25.343042 6.949155 25.699020 7.014423 26.055310 7.078895 26.411909 7.142570 26.766816 7.205447 27.126030 7.267524 27.483547 7.328801 27.841366 7.389276 28.199487 7.448948 28.557905 7.507817 28.916620 7.565881 29.275628 7.623138 29.634929 7.679589 29.994519 7.735231 30.354397 7.790063 30.714561 7.844085 31.075008 7.897296 31.435738 7.949694 31.796747 8.001279 32.158033 8.052049 32.519596 8.102003 32.881432 8.151140 33.243539 8.199460 33.605916 8.246961 33.968560 8.293642 34.331470 8.339502 34.694643 8.384541 35.058077 8.428756 35.421770 8.472148 35.785720 8.514715 36.149926 8.556457 36.514384 8.597372 36.879093 8.637459 37.244051 8.676718 37.609255 8.715147 37.974704 8.752745 38.340396 8.789513 38.706327 8.825448 39.072498 8.860550 39.438904 8.894817 39.805544 8.928250 40.172416 8.960846 40.539519 8.992606 40.906848 9.023528 41.274404 9.053612 41.642183 9.082856 42.010183 9.111260 42.378403 9.138823 42.746839 9.165543 43.115491 9.191421 43.484355 9.216455 43.853431 9.240645 44.222714 9.263989 44.592205 9.286458 44.961899 9.308139 45.331796 9.328943 45.701892 9.348898 46.072187 9.368003 46.442677 9.386259 46.813361 9.403664 47.184236 9.420217 47.555300 9.435918 47.926551 9.450766 48.297987 9.464759 48.669606 9.477899 49.041406 9.490183 49.413384 9.501610 49.785638 9.512181 50.157866 9.521895 50.530366 9.530751 50.903036 9.538747 51.275873 9.545884 51.648875 9.552161 52.022041 9.557577 52.395368 9.562131 52.768853 9.565823 53.142495 9.568652 53.516291 9.570618 53.890239 9.571719 54.264338 9.571956 54.638584 9.571327 55.012976 9.569833 55.387511 9.567471 55.762187 9.564243 56.137002 9.560146 56.511954 9.555181 56.887041 9.549347 57.262260 9.542644 57.637609 9.535070 58.013086 9.526625 58.388688 9.517310 58.764415 9.507122 59.140262 9.496062 59.516228 9.484130 59.892312 9.471323 60.268509 9.457643 60.644820 9.443089 61.021240 9.427659 61.397763 9.411354 61.774402 9.394173 62.151139 9.376116 62.527978 9.357182 62.904916 9.337370 63.281950 9.316681 63.659079 9.295113 64.036300 9.272667 64.413612 9.249341 64.791011 9.225136 65.168495 9.200050 65.546063 9.174085 65.923712 9.147238 66.301440 9.119510 66.679245 9.090900 67.057124 9.061409 67.435075 9.031035 67.813096 8.999778 68.191184 8.967638 68.569338 8.934614 68.947555 8.900706 69.325833 8.865914 69.704170 8.830238 70.082563 8.793676 70.461010 8.756229 70.839508 8.717897 71.218057 8.678678 71.596653 8.638574 71.975294 8.597583 72.353978 8.555705 72.732702 8.512939 73.111465 8.469287 73.490263 8.424747 73.869096 8.379318 74.247960 8.333002 74.626854 8.285797 75.005774 8.237704 75.384719 8.188721 75.763687 8.138849 76.142675 8.088088 76.521681 8.036438 76.900703 7.983897 77.279738 7.930467 77.658784 7.876146 78.037840 7.820934 78.416902 7.764833 78.795968 7.707840 79.175036 7.649956 79.554105 7.591181 79.933171 7.531515 80.312232 7.470958 80.691287 7.409509 81.070332 7.347168 81.449366 7.283935 81.828386 7.219811 82.207390 7.154794 82.586376 7.088885 82.965341 7.022084 83.344283 6.954391 83.723201 6.885804 84.102091 6.816326 84.480952 6.745955 84.859781 6.674691 85.238575 6.602534 85.617333 6.529485 85.996053 6.455542 86.374732 6.380707 86.753367 6.304979 87.131957 6.228358 87.510500 6.150843 87.888992 6.072436 88.267432 5.993136 88.645818 5.912943 89.024147 5.831857 89.402417 5.749877 89.780626 5.667005 90.158771 5.583240 90.536850 5.498582 90.914861 5.413031 91.292802 5.326588 91.670670 5.239251 92.048464 5.151022 92.426180 5.061900 92.803817 4.971886 93.181372 4.880979 93.558844 4.789180 93.336229 4.696488 94.313526 4.602904 94.690732 4.508428 95.067845 4.413060 95.444863 4.316801 95.821784 4.219649 96.198605 4.121606 96.575324 4.022671 96.951938 3.922846 97.328447 3.822128 97.704847 3.720520 98.081135 3.618021 98.457311 3.514632 98.833371 3.410352 99.209313 3.305181 99.585136 3.199121 99.960836 3.092170 100.336412 2.984330 100.711861 2.875600 101.087181 2.765981 101.462370 2.655473 101.837426 2.544076 102.212346 2.431791 102.587128 2.318617 102.961771 2.204555 103.336270 2.089605 103.710625 1.973768 104.084834 1.857043 104.458893 1.739431 104.832801 1.620933 105.206555 1.501548 105.580154 1.381277 105.953595 1.260120 106.326875 1.138077 106.699993 1.015149 107.072946 0.891337 107.445733 0.766639 107.818350 0.641058 108.190796 0.514592 108.563069 0.387243 108.935165 0.259011 109.307084 0.129896 30° ≦ θ ≦ 60° 0.371910 0.129871 0.744227 0.258937 1.116948 0.387199 1.490071 0.514655 1.863596 0.641303 2.237519 0.767143 2.611839 0.892174 2.986553 1.016395 3.361661 1.139804 3.737159 1.262401 4.113046 1.384184 4.489319 1.505153 4.865978 1.625307 5.243019 1.744643 5.620441 1.863163 5.998241 1.980864 6.376419 2.097745 6.754971 2.213805 7.133896 2.329045 7.513192 2.443461 7.892849 2.557052 8.272873 2.669819 8.653262 2.781760 9.034014 2.892875 9.415126 3.003162 9.796596 3.112621 10.178424 3.221251 10.560605 3.329051 10.943140 3.436020 11.326025 3.542157 11.709258 3.647461 12.092838 3.751931 12.476762 3.855566 12.861028 3.958365 13.245635 4.060329 13.630580 4.161454 14.015861 4.261742 14.401477 4.361190 14.787424 4.459798 15.173701 4.557565 15.560307 4.654490 15.947238 4.750573 16.334493 4.845812 16.722070 4.940207 17.109967 5.033756 17.498181 5.126460 17.886711 5.218317 18.275554 5.309326 18.664709 5.399487 19.054173 5.488798 19.443945 5.577259 19.834021 5.664870 20.224401 5.751628 20.615082 5.837535 21.006062 5.922588 21.397338 6.006786 21.788910 6.090131 22.180774 6.172619 22.572928 6.254252 22.965372 6.335027 23.358101 6.414945 23.751115 6.494004 24.144411 6.572203 24.537987 6.649543 24.931841 6.726022 25.325971 6.801640 25.720375 6.876395 26.115050 6.950288 26.509996 7.023317 26.905208 7.095482 27.300686 7.166782 27.696427 7.237216 28.092429 7.306784 28.488691 7.375485 28.885209 7.443319 29.281982 7.510284 29.679007 7.576380 30.076283 7.641607 30.473808 7.705964 30.871579 7.769450 31.269593 7.832064 31.667850 7.893806 32.066347 7.954676 32.465082 8.014672 32.864052 8.073795 33.263256 8.132042 33.662691 8.189415 34.062356 8.245912 34.462247 8.301533 34.862364 8.356277 35.262703 8.410144 35.663263 8.463133 36.064042 8.515243 36.465037 8.566474 36.866246 8.616825 37.267667 8.666297 37.669299 8.714887 38.071138 8.762597 38.473183 8.809424 38.875431 8.855370 39.277881 8.900432 39.680530 8.944612 40.083376 8.987907 40.486417 9.030319 40.889651 9.071845 41.293076 9.112486 41.696689 9.152242 42.100488 9.191111 42.504472 9.229094 42.908637 9.266190 43.312983 9.302398 43.717506 9.337718 44.122205 9.372149 44.527077 9.405692 44.932120 9.438345 45.337332 9.470109 45.742711 9.500983 46.148255 9.530965 46.553962 9.560057 46.959829 9.588258 47.365854 9.615567 47.772035 9.641983 48.178370 9.667507 48.584857 9.692138 48.991494 9.715876 49.398278 9.738720 49.805207 9.760671 50.212279 9.781726 50.619492 9.801887 51.026844 9.821153 51.434333 9.839524 51.841955 9.856998 52.249710 9.873577 52.657595 9.889259 53.065608 9.904045 53.473747 9.917933 53.882009 9.930925 54.290393 9.943018 54.698896 9.954214 55.107515 9.964511 55.516250 9.973910 55.925097 9.982410 56.334055 9.990012 56.743121 9.996714 57.152293 10.002516 57.561569 10.007418 57.970947 10.011421 58.380425 10.014523 58.790000 10.016725 59.199670 10.018025 59.609433 10.018425 60.019288 10.017924 60.429231 10.016521 60.839260 10.014217 61.249374 10.011011 61.659570 10.006903 62.069846 10.001892 62.480200 9.995979 62.890629 9.989164 63.301132 9.981446 63.711707 9.972825 64.122350 9.963300 64.533060 9.952873 64.943836 9.941542 65.354673 9.929308 65.765572 9.916170 66.176528 9.902128 66.587540 9.887182 66.998607 9.871332 67.409725 9.854578 67.820892 9.836919 68.232107 9.818357 68.643367 9.798889 69.054670 9.778517 69.466014 9.757241 69.877396 9.735059 70.288815 9.711973 70.700268 9.687982 71.111754 9.663086 71.523269 9.637284 71.934812 9.610578 72.346381 9.582966 72.757972 9.554450 73.169586 9.525027 73.581218 9.494700 73.992867 9.463467 74.404531 9.431329 74.816207 9.398286 75.227894 9.364337 75.639589 9.329483 76.051290 9.293723 76.462994 9.257058 76.874701 9.219488 77.286407 9.181012 77.698110 9.141631 78.109808 9.101344 78.521499 9.060152 78.933182 9.018055 79.344852 8.975053 79.756510 8.931145 80.168151 8.886333 80.579775 8.840615 80.991379 8.793993 81.402960 8.746465 81.814517 8.698032 82.226048 8.648695 82.637550 8.598453 83.049021 8.547307 83.460459 8.495255 83.871862 8.442300 84.283227 8.388440 84.694553 8.333676 85.105837 8.278008 85.517078 8.221436 85.928272 8.163960 86.339419 8.105580 86.750515 8.046297 87.161558 7.986110 87.572547 7.925020 87.983480 7.863027 88.394353 7.800131 88.805165 1.736332 89.215914 7.671630 89.626597 7.606026 90.037214 7.539520 90.447760 7.472112 90.858234 7.403801 91.268635 7.334589 91.678959 7.264476 92.089205 7.193461 92.499371 7.121545 92.909454 7.048728 93.319452 6.975011 93.729363 6.900393 94.139186 6.824875 94.548917 6.748457 94.958555 6.671139 95.368097 6.592922 95.777542 6.513805 96.186888 6.433790 96.596131 6.352876 97.005270 6.271063 97.414303 6.188352 97.823228 6.104744 98.232043 6.020237 98.640745 5.934834 99.049332 5.848533 99.457803 5.761336 99.866155 5.673243 100.274385 5.584253 100.682493 5.494367 101.090475 5.403586 101.498330 5.311910 101.906055 5.219339 102.313648 5.125874 102.721108 5.031514 103.128432 4.936261 103.535618 4.840114 103.942664 4.743074 104.349567 4.645142 104.756326 4.546317 105.162939 4.446600 105.569403 4.345991 105.975716 4.244492 106.381876 4.142101 106.787882 4.038820 107.193730 3.934649 107.599420 3.829588 108.004948 3.723638 108.410312 3.616799 108.815512 3.509072 109.220543 3.400456 109.625406 3.290954 110.030096 3.180563 110.434612 3.069287 110.838953 2.957124 111.243116 2.844075 111.647098 2.730141 112.050898 2.615321 112.454514 2.499618 112.857944 2.383030 113.261185 2.265559 113.664235 2.147205 114.067093 2.027968 114.469756 1.907849 114.872223 1.786849 115.274490 1.664967 115.676556 1.542205 116.078420 1.418563 116.480078 1.294041 116.881529 1.168640 117.282771 1.042361 117.683802 0.915203 118.084619 0.787168 118.485221 0.658256 118.885605 0.528468 119.285769 0.397804 119.685712 0.266264 120.085432 0.133850 120.484925 0.000561 60° ≦ θ ≦ 90° 0.400229 0.131263 0.800795 0.261688 1.201698 0.391276 1.602934 0.520024 2.004502 0.647934 2.406401 0.775002 2.808627 0.901230 3.211179 1.026617 3.614056 1.151160 4.017254 1.274861 4.420772 1.397717 4.824608 1.519729 5.228761 1.640895 5.633227 1.761215 6.038005 1.880689 6.443093 1.999314 6.848490 2.117092 7.254192 2.234021 7.660198 2.350099 8.066506 2.465328 8.473114 2.579706 8.880020 2.693232 9.287221 2.805905 9.694717 2.917726 10.102504 3.028693 10.510582 3.138805 10.918947 3.248063 11.327598 3.356465 11.736532 3.464010 12.145749 3.570699 12.555245 3.676530 12.965019 3.781503 13.375069 3.885617 13.785392 3.988871 14.195987 4.091266 14.606851 4.192800 15.017983 4.293473 15.429380 4.393284 15.841041 4.492232 16.252964 4.590317 16.665145 4.687539 17.077585 4.783896 17.490279 4.879389 17.903227 4.974016 18.316426 5.067778 18.729874 5.160673 19.143569 5.252701 19.557510 5.343861 19.971694 5.434153 20.386118 5.523577 20.800782 5.612131 21.215682 5.699816 21.630818 5.786630 22.046186 5.872573 22.461785 5.957646 22.877613 6.041846 23.293667 6.125174 23.709947 6.207629 24.126448 6.289211 24.543170 6.369919 24.960111 6.449753 25.377268 6.528712 25.794640 6.606795 26.212223 6.684003 26.630017 6.760335 27.048019 6.835790 27.466228 6.910368 27.884640 6.984068 28.303254 7.056891 28.722068 7.128835 29.141080 7.199899 29.560288 7.270085 29.979690 7.339391 30.399283 7.407816 30.819066 7.475361 31.239036 7.542025 31.659192 7.607807 32.079531 7.672708 32.500052 7.736726 32.920751 7.799862 33.341628 7.862114 33.762681 7.923484 34.183906 7.983969 34.605302 8.043570 35.026867 8.102287 35.448598 8.160118 35.870495 8.217065 36.292554 8.273125 36.714774 8.328300 37.137152 8.382588 37.559687 8.435990 37.982376 8.488505 38.405217 8.540132 38.828209 8.590872 39.251349 8.640723 39.674635 8.689686 40.098064 8.737761 40.521636 8.784947 40.945347 8.831243 41.369196 8.876650 41.793181 8.921167 42.217300 8.964794 42.641549 9.007530 43.065929 9.049376 43.490435 9.090330 43.915067 9.130394 44.339822 9.169566 44.764698 9.207846 45.189693 9.245234 45.614805 9.281730 46.040031 9.317333 46.465371 9.352044 46.890821 9.385861 47.316379 9.418785 47.742044 9.450816 48.167813 9.481953 48.593685 9.512197 49.019657 9.541546 49.445727 9.570000 49.871893 9.597561 50.298153 9.624226 50.724505 9.649997 51.150946 9.674872 51.577475 9.698852 52.004090 9.721937 52.430788 9.744126 52.857568 9.765419 53.284427 9.785817 53.711363 9.805318 54.138375 9.823923 54.565459 9.841631 54.992614 9.858443 55.419839 9.874358 55.847130 9.889376 56.274485 9.903497 56.701903 9.916721 57.129382 9.929048 57.556919 9.940478 57.984513 9.951010 58.412161 9.960644 58.839860 9.969381 59.267610 9.977220 59.695408 9.984161 60.123252 9.990204 60.551139 9.995349 60.979068 9.999596 61.407037 10.002945 61.835043 10.005395 62.263085 10.006947 62.691160 10.007601 63.119266 10.007356 63.547402 10.006213 63.975564 10.004171 64.403751 10.001231 64.831962 9.997392 65.260193 9.992654 65.688442 9.987018 66.116709 9.980483 66.544990 9.973049 66.973283 9.964717 67.401586 9.955486 67.829898 9.945356 68.258216 9.934327 68.686538 9.922409 69.114862 9.909574 69.543186 9.895849 69.971508 9.881226 70.399825 9.865704 70.828136 9.849283 71.256439 9.831964 71.684730 9.813746 72.113010 9.794630 72.541274 9.774615 72.969522 9.753702 73.397751 9.731890 73.825959 9.709181 74.254144 9.685573 74.682304 9.661067 75.110436 9.635663 75.538539 9.609360 75.966611 9.582160 76.394649 9.554063 76.822652 9.525067 77.250617 9.495174 77.678542 9.464384 78.106425 9.432696 78.534265 9.400111 78.962058 9.366628 79.389804 9.332249 79.817499 9.296973 80.245142 9.260800 80.672731 9.223730 81.100263 9.185764 81.527737 9.146902 81.955150 9.107143 82.382501 9.066489 82.809787 9.024939 83.237006 8.982493 83.664156 8.939151 84.091236 8.894914 84.518242 8.849782 84.945173 8.803755 85.372028 8.756833 85.798803 8.709017 86.225496 8.660306 86.652106 8.610702 87.078631 8.560203 87.505069 8.508810 87.931416 8.456524 88.357672 8.403345 88.783835 8.349272 89.209901 8.294307 89.635870 8.238449 90.061738 8.181699 90.487505 8.124056 90.913168 8.065522 91.338724 8.006096 91.764173 7.945779 92.189511 7.884570 92.614736 7.822471 93.039848 7.759482 93.464843 7.695602 93.889719 7.630832 94.314475 7.565172 94.739108 7.498623 95.163616 7.431185 95.587998 7.362858 96.012251 7.293642 96.436373 7.223539 96.860362 7.152547 97.284216 7.080668 97.707933 7.007902 98.131511 6.934249 98.554948 6.859709 98.978242 6.784283 99.401391 6.707971 99.824392 6.630774 100.247244 6.552691 100.669944 6.473724 101.092491 6.393872 101.514883 6.313136 101.937117 6.231517 102.359191 6.149014 102.781104 6.065628 103.202853 5.981359 103.624437 5.896208 104.045853 5.810176 104.467099 5.723262 104.888173 5.635467 105.309073 5.546792 105.729798 5.457236 106.150344 5.366801 106.570711 5.275486 106.990895 5.183292 107.410896 5.090220 107.830710 4.996270 108.250337 4.901442 108.669773 4.805737 109.089017 4.709155 109.508067 4.611698 109.926921 4.513364 110.345576 4.414155 110.764031 4.314071 111.182284 4.213112 111.600333 4.111280 112.018175 4.008574 112.435809 3.904996 112.853233 3.800545 113.270444 3.695222 113.687441 3.589027 114.104222 3.481961 114.520784 3.374025 114.937125 3.265219 115.353244 3.155544 115.769139 3.044999 116.184807 2.933587 116.600247 2.821306 117.015456 2.708158 117.430433 2.594143 117.845175 2.479262 118.259681 2.363516 118.673948 2.246904 119.087975 2.129427 119.501759 2.011087 119.915299 1.891883 120.328592 1.771816 120.741637 1.650887 121.154431 1.529096 121.566973 1.406444 121.979261 1.282931 122.391291 1.158559 122.803064 1.033327 123.214576 0.907236 123.625825 0.780287 124.036811 0.652481 124.447529 0.523817 124.857980 0.394298 125.268160 0.263922 125.678067 0.132692 126.087701 0.000607

A pump casing can be divided into convenient sizes and manufactured in sections. Finished parts can be assembled with nuts and bolts provided in the package, following instructions, to form a casing of the desired curvature.

FIG. 9 describes the disassembled parts of an eccentric rotor vane pump of horizontal design, and FIG. 10 describes the disassembled parts of an eccentric rotor vane pump of vertical design. FIG. 11 shows the components of the eccentric rotor vane pump described in this invention. In the manufacture of large-sized casings using the existing manufacturing method, the entire casing is manufactured as a single piece and the size of the rotor increases in proportion to the size of the casing. In this case the processing of the accurate guide surface which meets with the sliding, scraping vane is severely disabled. In order to overcome this limitation, two semi-circular rotors (5 and 5′) are separately manufactured, as shown in FIG. 11. On the inside of each semi-circular rotor, guide grooves (7′) are formed to match the projecting parts {circle around (7)} on both sides of vane {circle around (3)}, so that the projecting parts can move along the grooves when the vane slides back and forth. The casing parts (1 and 6) are held together with bolts and side covers (9 and 9′) are tightly placed on the open sides of the casing also using bolts. The rotating discs (8 and 8′) drives the eccentric rotor (2) to otates in close contact with the inner surface of the casing. The sealing parts (10 and 10′) are fitted inside the side covers (9 and 9′), and sealing liquid is applied to the contacting surfaces between the sealing parts and the rotating discs (8 and 8′) and shafts (12 and 12′). The bearing boxes (11 and 11′) are attached to the sealing parts using bolts, to support the rotating shafts (12 and 12′).

The reference number 13 denotes the fluid inlet and the number 14, the fluid outlet. The number 16, 17 and 18 in the figures refer to bolts and nuts provided in the package. The number 15 in FIG. 10 denotes the thrust bearing which is used to support the weight of an eccentric rotor oI vertical shaft.

In an eccentric rotor vane pump of vertical shaft as shown in FIG. 10, the rotor experiences increasing weight as casing size increases. In addition to the lower shaft and the bearing in the bearing box, therefore, a large-sized pump as an in-built thrust bearing to support the weight and thus allow smooth rotations regardless of the rotor weight. As casing size increases, weight of the vane also increases. For this reason, vane {circle around (3)} is designed to reciprocate horizontally, along the guide faces of the vertical axial rotor. So the vane can slide and scrape the inner surface of the casing in close contact, no matter how large casing size and vane weight may be. Friction and centrifugal force generated by the rotating vane of a large-sized pump can also be greatly reduced. The weight of vane {circle around (3)} still affects the horizontal movement of the vane, while due to horizontal rotations the two ends of the vane, sliding and scraping in contact with the curved surface of the casing, can no longer affect the gravitational pull on the vane. Therefore vane {circle around (3)} is designed to contain the appropriate number of convex parts (7), and the semi-circular rotors, the same number of grooves (7′) as convex parts. Or a suitable device such as beating is installed at the center of mass on the upper or bottom side of the vane, so as to absorb and reduce the weight of vane {circle around (3)}. As a result, the eccentric rotor vane pump of this design can undertake smooth horizontal movement, which is one of the major purports of this invention.

Springless eccentric rotor vane pumps (of either horizontal or vertical shaft) with cardiocle and expanded cardioid casings derived from Equations (5) and (6), as explained above, solve the limitations of, and problems posed by, current eccentric rotor vane pumps. Processing of large-size pumps is now possible with mathematical formation of casing curatures, hitherto regarded as impossible. In addition, as these pumps can perform more revolutions per unit time, pump size can be greatly reduced; pumps one-fifth the size of curtent large-size, large-output pumps can produce the same amounts of output. Moreover the achievement of exact mathematical descriptions of the cardiocle and expanded cardioid is opening a new chapter in pump technology in terms of analytic geometry.

The following section on ‘what is claimed’ merely suggests a few applications of this invention. Further changes or corrections are still possible, but these are conceptually part of the invention. 

What is claimed is:
 1. A method of manufacturing casing curved surfaces for eccentric rotor vane pumps, wherein the cardiocle curvature of the casing in a springless eccentric rotor vane pump can be represented over the range 0°≦θ≦180° as X²+Y² ={2{square root over (r+L (2R−r+L ))}+( R−r)sin θ−{square root over (R ²+L −(R−r+L )² +L cos²+L θ)}}², P=2{square root over (r+L (2R−r+L ))}+( R−r)sin θ−{square root over (R ²+L −(R−r+L )²+L cos²+L θ)} ${P = {2{a\left( {1 + {\left( {R - r} \right)\frac{\sin \quad \theta}{2a}} - \frac{\sqrt{R^{2} - {\left( {R - r} \right)^{2}\quad \cos^{2}\theta}}}{2a}} \right)}}},$

and X²+Y²=({square root over (R ²+L −(R−r+L )² +L cos²+L θ)}−(R−r)sin θ)², or P={square root over (R²+L −(R−r+L )²+L cos²+L θ)}−( R−r)sin θ, where X and Y are Cartesian coordinates, r denoites the radius of the rotor, R denotes the radius of the basic circle, θ denotes the rotation angle of the rotor or vane, and P is a polar coordinate, whereby the above equations being implemented in the precision manufacture of the curved surface of the casing in the eccentric rotor vane pump using CNC techniques.
 2. The method according to claim 1, wherein the equation for the expanded cardioid curvature of the casing over the range 0°≦θ≦360° can be written as ${{P = {2a\left\{ {1 + {\left( {R - r} \right)\frac{\sin \quad \theta}{2a}}} \right\}}},}\quad$

which can be directly applied for the manufacture of the curved surface of the casing in the eccentric rotor vane pump, using CNC techniques.
 3. The method according to claim 1 or 2, wherein the curved surface of the casing in the eccentric rotor vane pump is designed and manufactured in sections, which are then assembled.
 4. A method of machining casing curved surfaces for eccentric rotor vane pumps, wherein the cardiocle curvature of the casing in a springless eccentric rotor vane pump can be represented over the range 0°≦θ≦180° as X²+Y² ={2{square root over (r+L (2R−r+L ))}+( R−r)sin θ−{square root over (R ²+L −(R−r+L )² +L cos²+L θ)}}², P=2{square root over (r+L (2R−r+L ))}+( R−r)sin θ−{square root over (R ²+L −(R−r+L )²+L cos²+L θ)} ${P = {2{a\left( {1 + {\left( {R - r} \right)\frac{\sin \quad \theta}{2a}} - \frac{\sqrt{R^{2} - {\left( {R - r} \right)^{2}\quad \cos^{2}\theta}}}{2a}} \right)}}},$

and X²+Y²=({square root over (R ²+L −(R−r+L )² +L cos²+L θ)}−(R−r)sin θ)², or P={square root over (R²+L −(R−r+L )²+L cos²+L θ)}−( R−r)sin θ, where X and Y are Cartesian coordinates, r denotes the radius of the rotor, R denotes the radius of the basic circle, θ denotes the rotation angle of the rotor or vane, and P is a polar coordinate, the above quations being implemented in the precision manufacture of the curved surface of the casing in the eccentric rotor vane pump, using CNC techniques.
 5. The method according to claim 4, wherein the equation for the expanded cardioid curvature of the casing over the range 180°≦θ≦360° can be written as ${{P = {2a\left\{ {1 + {\left( {R - r} \right)\frac{\sin \quad \theta}{2a}}} \right\}}},}\quad$

which can be directly applied for the manufacture of the curved surface of the casing in the eccentric rotor vane pump, using CNC techniques. 